Control charts are part of the repertoire of statistical tools collectively named as SPC (Statistical Process Control) or SQC (Statistical Quality Control). They were first applied to an industrial context by Walter Shewhart who is regarded as the father of SQC or SPC.
Control charts provide a set of powerful statistical tools that can be effectively used for quantitative monitoring and control of processes. Clear understanding of the statistical concepts underlying control charts is essential for their correct use.
Control Chart as a Graphical Test of Hypothesis
Conceptually speaking a control chart is a graphical test of hypothesis which involves testing the following null and alternative hypothesis (the test is a two-tailed test).
X, X bar Chart
Control chart sets the two controls limits UCL and LCL at 3 sigma on right side and 3 sigma on left side of the mean respectively. In other words,
In test of hypothesis, alpha is generally kept at 5% level. Hence it is obvious that the alpha value used in control charts is very less compared to what is generally used in the test of hypothesis. It is also to be noted that alpha is fixed for control charts.
It is useful to recollect that alpha is also known as the Type I error (which is the probability of detecting a shift when there is none).
Keeping a low alpha ensures that the control chart doesn't throw up too many false alarms. False alarms need investigation and have a cost associated hence in that sense control charts reduce the cost associated with false alarms.
Power of Control Chart
The power of a control chart is expressed in the form of 1 - beta, which is the probability of detecting a shift when there is one.
It is useful to recollect that beta is also known as the Type II error (which is the probability of not detecting a shift where this is one).
The power of a control chart can be represented using an OC Curve (operating characteristic curve). For a control chart alpha is fixed (in the manner explained above) and beta varies depending on certain factors.
One of the most important factor is the sample size. Higher the sample size higher the power of the control chart. This explains the reason why the X bar - R is more powerful than an I - MR chart.
Outliers and Non-random Patterns in Control Charts
The rules of SPC related to outliers and non-random patterns are derived from the fact that such events have a very very low probability of occurrence given that only common causes are present. So if they occur then something must have gone wrong and there is a strong indication against only common causes being present. Here are some considerations to take note of:
Control charts can be used to obtain a clear indication about the statistical stability of a process. A process is said to be in a state of "statistical stability" if it operates only under a set of common causes. Common causes are also called as non-assignable or inherent causes. As compared to that, a process is not stable if there are certain special or assignable causes present.
Control charts can also be used to assess the capability of the process. Plotting lines for the Upper Specification Limit (USL) and Lower Specification Limit (LSL) along with the UCL and LCL lines can provide good idea regarding the capability of the process. If the lines of UCL and LCL are well within the lines of USL and LSL, the process can be called as a capable process.
In respect of stability and capability following points need special mention:
Control charts provide a set of powerful statistical tools that can be effectively used for quantitative monitoring and control of processes. Clear understanding of the statistical concepts underlying control charts is essential for their correct use.
Control Chart as a Graphical Test of Hypothesis
Conceptually speaking a control chart is a graphical test of hypothesis which involves testing the following null and alternative hypothesis (the test is a two-tailed test).
X, X bar Chart
- Ho : Mu = Mu 0 (there is no shift in process mean)
- H1 : Mu is not equal to M 0 (there is shift in process mean)
- Ho : Sigma ^ 2 = Sigma 0 ^2 (there is no shift in process variance)
- H1 : Sigma ^ 2 is not equal to Sigma 0 ^2 (there is shift in process variance)
- Upper Control Limit (UCL)
- Center Line (CL)
- Lower Control Limit (LCL)
Control chart sets the two controls limits UCL and LCL at 3 sigma on right side and 3 sigma on left side of the mean respectively. In other words,
- UCL = CL + 3 sigma
- LCL = CL - 3 sigma
- sigma = MR bar / d2
- where d2 is a statistical constant that depends on the sample size 'n'
- and for Individual and Moving Range chart, n =2
In test of hypothesis, alpha is generally kept at 5% level. Hence it is obvious that the alpha value used in control charts is very less compared to what is generally used in the test of hypothesis. It is also to be noted that alpha is fixed for control charts.
It is useful to recollect that alpha is also known as the Type I error (which is the probability of detecting a shift when there is none).
Keeping a low alpha ensures that the control chart doesn't throw up too many false alarms. False alarms need investigation and have a cost associated hence in that sense control charts reduce the cost associated with false alarms.
Power of Control Chart
The power of a control chart is expressed in the form of 1 - beta, which is the probability of detecting a shift when there is one.
It is useful to recollect that beta is also known as the Type II error (which is the probability of not detecting a shift where this is one).
The power of a control chart can be represented using an OC Curve (operating characteristic curve). For a control chart alpha is fixed (in the manner explained above) and beta varies depending on certain factors.
One of the most important factor is the sample size. Higher the sample size higher the power of the control chart. This explains the reason why the X bar - R is more powerful than an I - MR chart.
Outliers and Non-random Patterns in Control Charts
The rules of SPC related to outliers and non-random patterns are derived from the fact that such events have a very very low probability of occurrence given that only common causes are present. So if they occur then something must have gone wrong and there is a strong indication against only common causes being present. Here are some considerations to take note of:
- Any point outside the 3 sigma limits on either side is considered as an outlier. The reason - probability of such an event occurring is less than 0.27%
- Any trend or pattern is unlikely to form if the process is a stable normal distribution. In a stable normal distribution the mean and sigma remain constant and individual observations fall randomly as governed by the probability density function of the normal distribution
Control charts can be used to obtain a clear indication about the statistical stability of a process. A process is said to be in a state of "statistical stability" if it operates only under a set of common causes. Common causes are also called as non-assignable or inherent causes. As compared to that, a process is not stable if there are certain special or assignable causes present.
Control charts can also be used to assess the capability of the process. Plotting lines for the Upper Specification Limit (USL) and Lower Specification Limit (LSL) along with the UCL and LCL lines can provide good idea regarding the capability of the process. If the lines of UCL and LCL are well within the lines of USL and LSL, the process can be called as a capable process.
In respect of stability and capability following points need special mention:
- A process which is not stable may still be capable. This can happen in following cases:
- when the specification limits are set very wide apart and the inherent variation due to the design of the existing process is in a much narrower zone (this could be case of over-design and may mean high cost was incurred in setting the process)
- when the process is centered close to the mean, has an extremely low variation and constantly experiences minor shifts in the process mean.
- A process which is stable may not be capable. This can happen in following cases:
- when the process is centered close to the mean, has a large variation and constantly experiences shifts, minor or major, in the process mean
- when the process is centered away from the mean, has some variation, either small or large, and constantly experiences shifts, minor or major, in the process mean.
- A process may be stable around a certain value of the mean and then undergo a shift and become stable around another value of the mean. It might appear that the process has large variation in this case but in reality the variation around a certain mean is not that large.
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